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Linear Algebra Test
(1) Determine whether each matrix below is in echelon form, reduced echelon form, or neither of these.
(2) Given the system of equations
(a) Write down the corresponding augmented matrix
(b) Show explicitly the first few (around three) row operations . Stop when the
matrix is in echelon form.
(c) How many pivots will this matrix have?
(3) An system of linear equations and the corresponding rowreduced echelon form matrix appear below:
(a) Write down the general solution in the homogeneous case
(when b_{1} = b_{2} = b_{3} = 0)
(b) Is there a solution when b _{1} = 3, b_{2} = 1, and b_{3} = 4 ?
(c) What must the relation be among the quantities b_{1}, b_{2}, and b_{3} in order
for this system of equations to be consistent ?
(4) True or False?
(a) If a system of linear equations has two different solutions, then it must
have infinitely many solutions.
(b) The equation has only the trivial
solution if and only if the rowreduced
matrix has a pivot in each row.
(5) For which value of h is the vector in the set span
(6) Determine whether each set of vectors below is linearly independent or
linearly
dependent:
(7) If a linear transformation is defined by
(a) What is the domain of T?
(b) What is the codomain of T?
(c) Is the column vector [1, 1, 1] in the range of T ?
(d) If the column vector is in the range of
T, what relation must
hold between the quantities b_{1} , b_{2} , and b_{3} ?
(e) Describe geometrically the range of the transformation T.
(f) Is the transformation T onto ?
(g) Is the transformation T onetoone ?
(8) True or False?
(a) If is a linearly dependent set in R^{3},
then each vector is a
linear combination of the other two.
(b) If , then T is not a linear transformation.
(9) Find the standard matrix of the linear transformation
T : R^{2} > R^{2} which
first reflects points across the diagonal line x_{2} = x_{1} , and then reflects
across
the horizontal axis x_{2} = 0.
(10) Write down a reduced echelon matrix for an augmented
matrix which represents
an overdetermined system of 5 homogeneous linear equations in 3 unknowns
with one free variable .
(11) Suppose that the matrix A on the left row reduces to
the matrix B to the
right
Denote the columns of A by , and the columns of B by
(a) True or False: The solution set of
is identical to the solution set
of
(b) Is
(c) Is span
(d) Do the columns of A form a linearly independent set?
(e) Is the set a linearly independent set?
(f) Is the linear transformation onto?
onetoone ?
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